A critical case of stability in a free boundary problem (Q5930662)

The paper deals with the stability of the planar travelling wave solution for the following free boundary problem \[ \begin{gathered} u_t=u_{xx}+u_{yy} \quad \text{in } \Omega _t=\{ (x,y)\in \mathbb{R}^2:x\leq s(t,y)\},\;t>0, \\ u=u_n=1 \quad \text{on }\partial\Omega _t=\{ (x,y)\in \mathbb{R}^2:x= s(t,y)\},\;t>0, \end{gathered} \] here \(n\) is the normal vector to the free boundary \( \partial\Omega _t \) with positive \(x-\)component. The restriction to the space dimension two is only for convenience but not essential.